Suppose we have an oriented graph and max-flow has been determined.
I found that to determine min-cut or minimum s-t cut can then be found by labeling graph nodes such that nodes belonging to source are those which can be reached from source by unsaturated edges. On the contrary: Nodes belonging to sink are all the other, i.e. nodes reachable from sink by saturated edges.
My question is - why? I am unable to grasp this "phenomenon" intuitively.
In other words: Why the minimum cut cannot go through any other egdes, but those who connect nodes label as "belonging to the source" and those "belonging to the sink". Why the min-cut does not go through any other nodes?